LEP Data and the Stability of the Potential Confront the mSUGRA Model

The requirement that the supersymmetric scalar potential be stable in the minimal supergravity (mSUGRA) model imposes an upper bound on the universal gaugino mass $m_{1/2}$ as function of the common scalar mass $m_0$. Combining this with the experimental lower bound on $m_{1/2}$ from LEP data, we find a new lower bound on $m_0$, stronger than the one that comes from experimental data alone. If the corresponding upper and lower limits on the superparticle masses, presented in this letter, are found to be violated at Tevatron Run II or at the LHC, it would imply that we are living on a false vacuum. Special attention has been paid in estimating the uncertainties in these predictions due to the choice of the renormalization scale. The implications of our limits for the constraints obtained by indirect methods(SUSY dark matter, g - 2 of the muon, $ b \to s \gamma$....) are briefly discussed.

supergravity (mSUGRA) model [2] with radiative electroweak symmetry breaking [3]. This model has only five free parameters. They are the common scalar mass (m 0 ), the common gaugino mass (m 1/2 ), the common trilinear coupling (A 0 ), the ratio of vacuum expectation values of two Higgs field (tan β) and the sign of µ, the higgsino mass parameter.
The mSUGRA model has been confronted with the data from LEP as well as from Run I of the Tevatron collider. Such efforts have resulted in some useful lower bounds on these parameters most notably on m 0 and m 1/2 [4,5,6,7]. In this paper we shall be concerned mainly with the limits obtained by the ALEPH collaboration on the mSUGRA parameter space [4]. The results obtained by the other LEP groups are similar but some of them do not strictly follow the mSUGRA scenario.
Unfortunately mSUGRA does not predict quantitative upper bounds on these parameters, which could make this model falsifiable in the near future. Of course there are upper bounds based on naturalness arguments [8] . These bounds, however, crucially depends on the value of the fine tuning parameter which, though intuitively appealing, is rather difficult to quantify.
It is, therefore, rather tempting to reexamine the constraints obtained from the stability of the supersymmetric scalar potential [9,10,11]. It has been demonstrated in the past that these constraints leads to upper bounds on m 1/2 as a function of m 0 [10,11]. Useful constraints also emerge within the framework of anomally mediated supersymmetry breaking and other models [12].
Admittedly these bounds can be evaded by assuming that we live in a false vacuum with a life time larger than the age of the universe [13]. Yet such bounds are important. If they are found to be violated after the discovery of SUSY be it at Tevatron Run II or at the LHC, it must be accepted within the mSGRA scenario that the universe is indeed built on a false vacuum.
The unbounded from below (UFB) [10] constraint imposes an upper limit on m 1/2 (denoted by m max 1/2 ) as function of m 0 (to be explained below). Combining this with the experimental lower bound denoted by m min 1/2 from LEP data [4], we find a strong lower bound on m 0 (denoted by m min 0 ) stronger than the one that comes from experimental data alone. This can be translated into a stronger lower bound on slepton mass. Using the lower bound (from experimental data) and upper bound (from the UFB bound) on m 1/2 for a given m 0 we derive upper and lower bounds on various super particle masses for fixed slepton masses.
This prediction can be tested in the next round of experiments.
To make this letter self contained, we briefly discuss the most important, model independent unbounded from below -3 (UFB3) constraint obtained by considering a certain direction in the field space [10]. The scalar potential in this direction is given by where g ′ and g are normalised gauge couplings of U(1) and SU (2) respectively, H u and H d are the neutral components of the two Higgs doublets, L i and E j are the scalar partners of the leptons belonging to the SU(2) L doublet and singlet respectively, λ E j is a Yukawa coupling and i, j are generation indices. Here i = j.
Note that we could substitute squarks for sleptons in Eq. 1, in which case i = j is allowed. The constraints on the parameter space arise from the requirement where V min 0 is the electroweak symmetry breaking minimum of the scalar potential evaluated at the SUSY breaking scale Q =M S and the scaleQ is chosen to bê where M(φ) max is the largest eigen value of the field dependent mass matrices. The VEVs |E j | and |L i | are determined by H u and some model parameters. The relevant equations can be found in [10]. The UFB3 potential in Eq. 1 is derived from the tree level scalar potential. It is well known that loop corrections to the potential may have important effects (e.g., the predictions from the one loop corrected potential has a reasonably mild scale dependence) [14]. The above choice of Q is designed to minimize the magnitude of the loop corrections to the potential which are At this scale, therefore, the tree level potential by itself gives fairly reliable results [10,14]. The price that one has to pay for this simplication is a slight complication in the choice of the scale. While studying V UFB3 as a function of H u , one must select the scale at each step since it is correlated with H u .
It should be emphasized that the above prescription only gives an order of magnitude estimate of the scaleQ for a given H u . In order to carry out practical computations leading to constraints on the parameter space,Q is set exactly equal to the maximum of the quantities given in the parenthesis. In the rest of the paper this scale will be referred to as the approximate scaleQ A (H u ). This approximation introduces an element of uncertainty in the derived constraints, which will be discussed in detail before the numerical results are presented.
As can be seen from Eq. (1), the regions of the parameter space, where m 2 Hu becomes large negative at the required scaleQ A (H u ), tends to violate the UFB3 condition (inequality 2). This is because the first term of Eq. (1) which dominates for moderate and large values of |H u |, could be negative in this case. However, the second term in (1), which is positive definite, may become competitive in certain cases (e.g., for j = 1, when the Yukawa coupling in the denominator is small), and a dangerous minimum may be avoided.
The UFB3 potential with sleptons (Eq. 1) was found to yield the strongest constraint among all the UFB and charge colour breaking (CCB) conditions in the low tan β case [10].
The results were generalized for large tan β scenarios in [11]. In order to get the optimum result, one has to take the largest λ E j in the second term of Eq. (1), which leads to the choice E j =τ R . Now the restriction i = j requires L i =ẽ L orμ L and excludes the choicẽ τ L . In the low tan β case this restriction, however, is of little consequence since all the left sleptons are degenerate to a very good approximation.
To find the points in the mSUGRA parameter space which violate the UFB3 constraint (inequality 2), we vary H u from the grand unification scale (M G ) to M Z . For each H u and a chosen set of the five input parameters the approximate scaleQ A (H u ), described after inequality 2, is found by an iterative method ( see [11] for the details of the procedure).
Next we evalute the function in Eq. 1. At large H u (i.e., also at largeQ A (H u ) ), the change of m 2 Hu from its boundary value m 2 0 at M G is negligible. Consequently the first term of Eq. 1, which dominates, is positive and the inequality is satisfied. As H u is decreased m 2 Hu becomes negative. Further lowering of H u may eventually make the first term sufficiently negative, leading to the violation of inequality 2 for the chosen set of parameters. This set is then excluded. For some sets of parameters V UFB3 is found to be violated for a very narrow range is chosen on the basis of an order of magnitude estimate one may wonder whether this choice is the true value of the scale at which conclusions based on the tree level potential are reliable. For some other sets of parameters, however, V UFB3 remains approximately flat for a range of values of H u ( orQ A (H u ) ) and ineqality 2 is violated for the entire range. The corresponding parameter set can be ruled out with more confidence. For H u beyond this range V UFB3 rises above V min 0 . It may be inferred that for this range ofQ A (H u ) values higher order effects are indeed small, the tree level UFB3 potential by itself is fairly scale independent and conclusions based on it are reliable.
The variation of V UFB3 with log 10 (Q A (H u )/GeV) is illustrated in Fig. 1 In view of the above discussions, we have introduced the following prescription to estimate the uncertainties in the limits derived in this letter due to the choice ofQ A (H u ). In the process of obtaining the upper bound on m 1/2 for a given m 0 , m 1/2 is increased starting from the LEP lower limit [4]. At a certain m 1/2 (= m max 1/2 ) the UFB3 condition is violated for the first time at a scaleQ A (H u ). Usually this first violation occurs for a very narrow range of Q A (H u ) values. We shall refer to this limit as the optimistic limit (OL). Any m 1/2 above this value violates the UFB3 condition for a larger range ofQ A (H u ) values. This is so because for a given set of m 0 , A 0 , tan β and sign of µ, a larger m 1/2 drives m 2 Hu in Eq. 1 to larger negative values [11]. Thus, e.g., the UFB3 constraint for a larger m 1/2 will also be violated at a relatively higherQ A (H u ). The conservative limit (CL) refers to that m 1/2 for which the UFB3 condition is violated for a range ofQ This choice of the range is guided by the conventional understanding of the order of magnitude.
In Fig. 2 we present the allowed region of the m 0 − m 1/2 plane for tan β = 15, A 0 = 0 and µ > 0. In this figure as well as in the following ones, we have chosen the mSUGRA parameters as in ref [4], so that we can combine our theoretical constraints with the experimental limits without any ambiguity. Region I is excluded by the LEP lower limit on the light Higgs mass (m h ), which leads to the strongest experimental constraint in the m 0 − m 1/2 plane in the mSUGRA model. We have used the model independent limits on m h as a function of the mixing parameters obtained by the ALEPH collaboration (see Fig 4 of ref [15]) to reconstruct the upper edge of region I, which represents the lower limits on m 1/2 for different m 0 values. Region IV is excluded by the requirement that the lightest neutralino be the LSP which leads to an upper bound on m 1/2 for a given m 0 . Thus region II and III represent the allowed parameter space (APS) after imposing the experimental and the neutralino LSP constraints.
The APS in Fig. 2 is in good agreement with the results of ALEPH collaboration [4], as can be seen by comparing region I of Fig. 2 with the corresponding region of [4] obtained for the same set of mSUGRA parameters. We have used ISAJET version 7.64 [16] for computing the renormalization group evolution of the mSUGRA parameters and the resulting sparticle spectrum. Since m h is the most important parameter in constraining the m 0 −m 1/2 parameter space, we have also used the FeynHiggs programm [17] to compute the Higgs masses. In the following we shall show that the resulting discrepancy is not very serious by comparing the computed m h values from the two programs.
We now impose the UFB3 constraint (inequality 2) and obtain the optimistic upper limit The APS for other choices of tan β and sign of µ are presented in Fig. 3 and 4. In each case we see that the effectiveness of the theoretical and experimental constraints acting in tandem is much better than any one of them operating singly. The lower limit on m 0 (m min 0 ) in each case is significantly stronger than that obtained from the data alone.
We present in Fig. 5, which is a blown up version of the APS in Fig. 2, the uncertainty in m max 1/2 due to the choice of scale for tan β = 15, A 0 = 0, µ > 0. The thick lines represent information already present in Fig. 2. The optimistic limits (OLs) as defined above are represented by the thick dash-dotted line. The conservative limits (CLs) are given by the line 'a'. Due to the upward shift of the limits m min 0 s also get relaxed. It is to be noted that the differences between the OLs and the CLs become more prominent at larger value of m 0 .
In Table 1 Table 1. We see that the changes are < ∼ 10% in all cases.
Using the numbers in Table 2, we predict in Table 3 the lower limits on the slepton masses belonging to the first two generations. For low values of tan β, even the weaker conservative bounds derived using the UFB3 condition are significanly larger the corresponding results derived from m h and neutralino LSP constraints only. Detection of sleptons with masses below the above ranges in future experiments would directly signal the existence of minima deeper than the EW symmetry breaking vacuum. The chargino masses corresponding the conservative estimate of the lowest slepton mass are also presented in Table 3. Thus even if sleptons with masses in the ranges shown in Table 3 are found in future experiments, but the lighter chargino mass turns out to be appreciably larger , it would also indicate a violation of the UFB3 bound.
Similar lower limits can also be obtained for the squarks belonging to the first two generations. In mSUGRA the squark mass squared is given by( apart from the relatively unimportant D-term contributions) m 2 0 + cm 2 1/2 , where the dimensionless coefficient c lies between 6.0 and 6.5 [18] for different types of squarks. It is clear that the main contributions to such limits for relatively low m 0 values come from the experimental lower limits on m 1/2 . The stroger m min 0 as determined from the UFB3 condition, will in principle predict a lower bound stronger than the experimental bound . But it may be difficult to disitnguish between the two. The shift in mass limits due to the UFB condition, however, may play an important role once precision measurements of these masses are available.
We next discuss the predictions of the UFB3 constraint for slepton masses higher than the lowest allowed values presented in Table 3. For a fixed slepton mass both upper and lower bounds on several sparticle masses can be computed from the allowed ranges of m 1/2 presented in Figs. 2 -4. As an illustration we present in Fig. 5 the contour of left slepton mass of 300 GeV and 400 GeV. From the points of intersection of contours 'e' and 'f' with the lines representing the UFB3 bound and the experimental bound on m 1/2 , we predict in Table 4 the conservative and optimistic upper bounds and lower bounds on several sparticle masses. It is gratifying to note that the uncertainties in the upper limits due to the choice of scale is always < ∼ 10%. If gluinos happen to be the first sparticles to be discovered with a mass 700 GeV, say, it would predict that the lower bound on the slepton mass to be 265 GeV, as can be seen from line 'c' of Fig 5. The upper bounds in Table 4 indicate that the sparticle spectrum predicted by the mSUGRA model along with the reqirement of the stability of the scalar potential can indeed be tested in future experiments. However, it is desirable that the theoretical predictions, the uncertainties due to the choice of renormalization scale in particular, be further sharpened.
In this paper we have followed refs. [14] and [10] and assumed that the approximate scale, at which correct conclusions can be drawn from the tree level scalar potential, is equal to M(φ) max which is the dominant eigen value of the field dependent mass matrices. Our conservative limits on SUSY parameters are then obtained if these limits happen to be stable with respect to the variation of the scale within one order of magnitude of the above value.
In practice, however, the above eigen values may have a hierarchy. A more rigorous method [19] which takes into account this hiearchy and chooses the scale in a step by step fashion using the decoupling therem [20] is called for. A refined estimation of the scale will be particularly important if the measured masses are found to be in conflict with the theoretical upper bounds after the discovery of SUSY.
Interesting predictions about some decay properties of sparticles also follow from Fig 5. For example, below the curve marked 'b' the lighter chargino is the next lightest superparticle (NLSP). Thus a chargino of mass < ∼ 160 GeV is predicted to be the NLSP, since its contour marked 'd' in Fig. 5 lies below 'b' in the entire UFB3 allowed region. Consequently charginos with mass in this range can decay only via modes consisting of the LSP and SM particles.
In order to see whether the stability of the potential yields an improved absolute lower bound on the slepton masses compared to the experimental result, a more through scan of the parameter space is needed. This is beyond the scope of the present paper. However, following comments can be made.
Since the main objective of this letter is to combine the limits of [4] with the theoretical bounds obtained from the UFB3 condition, we restricted ourselves, as in [4] to A 0 = 0. For A 0 < 0, UFB3 bounds are more restrictive since m 2 Hu becomes more negative [11]. Indirect constraints on the m 0 -m 1/2 parameter space have been obtained [21,22] from the requirements that i) the prediction of the mSUGRA model be consistent with the dark matter density of the universe as given by the latest WMAP data [23], ii) it removes the alleged discrepancy between the measured value of the (g -2) of the muon [24] and the standard model prediction or iii) the prediction for the branching ratio of the inclusive process b → sγ be consistent with the measured value [25]. Cerdeno et al. [22] have also considered the impact of the UFB3 constraint on the parameter space. Our conclusions qualitatively agree with theirs.
For definiteness we compare the constaints obtained by Ellis et al [21] based on the latest WMAP data [23] with our constraints. A precise comparison is difficult since the constraints in [21] are given for different sets of SUSY parameters. Yet the following qualitative remarks can be made.
From the relevant constrained parameter space for tan β = 10, A 0 = 0 and both signs of µ can be found in Fig. (1a) and (1b) of [21]. Since our constraints are practically insensitive to the sign of µ for relatively small tan β, the constrained parameter space of Fig. 2 of this letter can be qualitatively compared with the results of [21]. It is interesting to note that for tan β = 10 the dark matter and the m h constraints allow only a very narrow region for low m 0 (∼ 100 GeV) and relatively large m 1/2 ( > ∼ 300 GeV). This is very likely to be in conflict with the UFB3 constraint. Similarly for tan β = 35, µ < 0 and A 0 = 0, there is a tiny region allowed by both dark matter matter and b → sγ constraints corresponding to m 0 between 250 -350 GeV and m 1/2 > 700 GeV. For tan β = 35 the UFB constraints are expected to be stronger than those presented in Fig. 3. Qualitatively one can estimate that the above regions will be disfavoured by the UFB3 constraints. We cannot comment on the compatibility of the large m 0 regions allowed by indirect constraints with the UFB3 condition without a fresh computation. The incompatibility between the post WMAP dark matter constraints and the UFB3 bound for relatively low tan β was also noted in [22] and we agree with them. On the basis of this observation tan β < ∼ 20 was ruled out by the above authors.
We note that such conclusions can be evaded by introducing a tiny R-parity violation which leaves the LSP essestially stable for collider experiments but unstable cosmologically. Such small effects can be naturally induced by higher dimension operators suppressed by a heavy mass scale. Thus values of tan β less than 20 are still very much relevant for the analysis of collider data.
Qualitatively one can also conclude that significant fractions of the UFB3 allowed pa-rameter space, especially for relatively low m 0 , will be disfavoured by the b → sγ constraint for all tan β and µ < 0. For µ > 0, on the other hand, the constraint from b → sγ is rather weak. In this scenario the region favoured by the g-2 constraint has significant overlap with the UFB3 allowed parameter space for all tan β.
Acknowlwdgements : A part of this work was done when AD was visiting the university of Dortmund under the follow-up visit programme of the Alexander von Humboldt Foundation. He thanks Professor E. A. Paschos for hospitatlity.